Lowcomplexity lattice reduction algorithm for MIMO detectors with tree searching
 Hyunsub Kim^{1}View ORCID ID profile,
 Hyukyeon Lee^{1},
 Jihye Koo^{1} and
 Jaeseok Kim^{1}Email author
https://doi.org/10.1186/s1363801607564
© The Author(s) 2016
Received: 22 May 2016
Accepted: 14 October 2016
Published: 18 January 2017
Abstract
In this paper, we propose a lowcomplexity lattice reduction (LR) algorithm for multipleinput multipleoutput (MIMO) detectors with tree searching. Whereas conventional approaches are based exclusively on channel characteristics, we focus on joint optimisation by employing an early termination criterion in the context of MIMO detection. In this regard, incremental LR (ILR) was previously proposed. However, the ILR is limited to LRaided successive interference cancellation (SIC) detectors which have considerable biterrorrate (BER) performance degradation compared to optimal detectors. Hence, in this paper, we extend the conventional ILR to be applicable to the LRaided detectors with nearoptimal performance. Furthermore, we perform the hypothetical analysis and several novel modifications to handle the obstacles for the application of the ILR to LRaided detectors other than the LRaided SIC detectors. The simulation results demonstrate that the computational complexity is considerably reduced, with BER performance degradation of 10^{−5}.
Keywords
1 Introduction
In an effort to satisfy the demand for highcapacity wireless communication systems, ample research is currently dedicated to multipleinput multipleoutput (MIMO) techniques, owing to their ability to provide diversity and multiplexing gain within limited bandwidth and power resources. However, a major obstacle to the realization of an enhanced MIMO system is the high computational complexity of its receiver. The optimal MIMO receiver is the maximumlikelihood (ML) detector (MLD). However, its complexity increases exponentially with the number of transmit antennas, making it infeasible for actual systems. Hence, a lowcomplexity design for MIMO receivers is a challenging research topic.
The sphere detector (SD) was introduced to achieve an optimal performance with low complexity, by employing a treesearching algorithm [1, 2]. However, the variable complexity of the SD is a major drawback for practical systems that require data to be processed at a constant rate. To overcome this drawback, the fixedcomplexity SD (FSD) [3, 4] and the Kbest detector [5] have been developed. These detectors have the advantage of constant throughput, because there is no feedback in the data flow. In particular, the FSD approaches optimal performance in a fixed number of operations. Nevertheless, the complexity of these algorithms remains high in MIMO systems with a large number of transmit antennas and higher order modulation.
Recently, lattice reduction (LR)aided detection methods have emerged as an efficient solution to the MIMO symboldetection problem [6]. LRaided linear and successive interference cancellation (SIC) detectors [7] provide the same diversity order as the ML detector, by transforming the system model with nearorthogonal channel matrices [8, 9]. Furthermore, the LRaided Kbest detector employs the Schnorr–Euchner enumeration to find the next child during tree searching [10–12]. However, a considerable gap remains between the performance of the optimal detector and those of conventional LRaided detectors as the number of transmit antennas increases. In this regard, an LRaided FSD has been developed to achieve nearML performance, despite a large number of antennas and higher order modulation [13–15].
Given these desirable features, research into LRaided detection has remained active. The majority of research on LRaided detection has been directed at a lowcomplexity LR algorithm. In [16], the complexvalued extension of the Lenstra–Lenstra–Lovász (LLL) [17] algorithm was proposed to reduce by half the size of the realvalued channel matrix. In [18], an effective LLL was proposed to reduce the complexity of the size reduction by processing only pairs of consecutive basis vectors. Moreover, some researchers focused on relaxing the Lovász condition to reduce complexity [19, 20]. Some effort in this field is devoted to fixing the complexity of the LLL algorithm. After the fixedcomplexity LLL (fcLLL) algorithm was first proposed in [21], its complexity was further reduced by modifying the column traverse strategy [22–24].
Whereas these approaches focus exclusively on channel characteristics, another approach is to jointly optimise LR processing and the detection process. The approach is referred to as incremental LR (ILR) [25]. ILR performs partial SIC detection at each iteration and employs an early termination (ET) criterion based on the reliability assessment (RA) [26] computed with the partial detection result. However, ILR cannot be applied to LRaided detectors other than the LRaided SIC detector.
In this paper, we propose a lowcomplexity LR algorithm with ET that can be employed to highperformance LRaided FSDs. In order to obtain the partial detection result in a simple manner, the proposed algorithm modifies the index of the column vectors for the channel matrix. Furthermore, in order to ensure that the characteristics of the latticereduced channel matrix remain the same, LR processing is performed only on the column vectors that involve LRaided detection, and a modified QR decomposition (QRD) is proposed to generate the partially upper triangular matrix. The experimental results demonstrate that the proposed method achieves a significant reduction in complexity while maintaining a performance degradation of less than 0.5 dB at a bit error rate (BER) of 10^{−5} for a 8×8 MIMO system with 256 QAM.
Notations: Uppercase and lowercase boldface letters are used for matrices and vectors, respectively. The superscripts (·)^{ T } and (·)^{ H } denote the transpose and the Hermitage of a matrix, respectively. a denotes the absolute value of a scalar a, or the cardinality of a if a is a set. ∥·∥ and ⌈·⌋ represent the 2norm of a vector and the rounding operation, respectively. I _{ N } denotes the N×N identity matrix, and 0 _{ M×N } denotes an M×N matrix of all zeros.
2 Preliminaries
In the following subsections, we briefly explain the MIMO system model and introduce the latticereduced MIMO system model, whereby the channel matrix is transformed so that it has a favorable characteristic for MIMO detection. Moreover, we introduce the FSD—and the LRaided FSD—which achieves nearoptimal performance with low complexity.
2.1 MIMO system model
where Ω denotes the constellation points. Whereas this MLD is optimal, its search space is proportional to \(\Omega ^{N_{T}}\phantom {\dot {i}\!}\). This exponential complexity renders the MLD infeasible for practical systems.
However, the noise amplification in (3) is the major cause of the degraded BER performance in linear detectors.
2.2 Latticereduced MIMO system model
where \(\dot {\mathbf {n}} = \frac {1}{\alpha }\mathbf {n}\).
Note that \(\mathbf {w} := \tilde {\mathbf {H}}^{\dagger } \dot {\mathbf {n}}\) in (9) is the noise amplified by the latticereduced channel matrix \(\tilde {\mathbf {H}}^{\dagger }\). With the aid of the nearorthogonal nature of the latticereduced matrix, the noise amplification in (9) is much less than (3) so that the BER performance of LRaided linear detectors has the same diversity order as the MLD.
2.3 FSD
where q _{ i } and r _{ i,j } are the ith element in q and the (i,j)th element in R, respectively. In order to achieve optimal performance, the channel matrix should be ordered prior to tree searching so that the signals with maximum and minimum postprocessing noise amplifications are detected at the FE and SE stages, respectively [3]. Throughout this paper, to simplify the notation, we consider the channel matrices and transmit signals as corresponding variables with permutations.
2.4 LRaided FSD [15]
In order to reduce the complexity, the LRaided FSD algorithm lowers the number of tree levels in the FE stage, in order to reduce the parent nodes generated at the FE stage. Nevertheless, the proposed algorithm maintains nearoptimal BER performance by adopting LRaided SIC at the SE stage. However, the application of the LR algorithm to the FSD is not straightforward. The bases of the channel matrix are modified in the latticereduced system model so that the child nodes cannot be expanded [10]. Hence, the LRaided FSD generates the candidate signal in the FE stage, and cancel the FE signals in the original constellation domain before transforming the system model into a latticereduced one.
where H ^{′} and s(k)′ respectively denote the N _{ R }×(N _{ T }−N _{ p }) channel matrix and the (N _{ T }−N _{ p })×1 transmitted signal whose lth (l=N _{ T }−N _{ p }+1,⋯,N _{ T }) column vectors and elements are nulled. Then, LRaided SIC is performed on (16) to complete the candidate list, and the detection is completed by the ML test, where the symbol vector with the minimum Euclidean distance (ED) is detected as the solution.
3 Proposed algorithm
3.1 Motivation
where A is the positive parameter that determines the tradeoff between performance and computational complexity. However, there is a critical issue in the application of the ET to the LRaided FSD. Whereas ILR obtained the practical ET criterion with RA by performing partial SIC detection during the intermediate LR process, all parent nodes should be considered as the candidate signal in the LRaided FSD, making the overhead incurred by the partial detection too large. A novel way of handling this obstacle is proposed in the next subsection.
3.2 CLLL with ET for the LRaided FSD
A problem arises when applying conventional ILR directly to the LRaided FSD [15]: all the parent nodes in the FE stage should be considered for partial detection, making the overhead incurred by the partial detection too large. In order to solve this problem, the proposed algorithm exchanges the column vectors of the channel matrix corresponding to the FE stage and SE stage. LR is performed on column vectors during the SE stage exclusively, and not during the FE stage. In this way, as the LRaided FSD performs LRaided detection during the SE stage, the column vectors that actually involve LRaided detection are latticereduced. Furthermore, because the signals of the SE stage are switched to the upper level of the tree structure, partial detection with SIC can be performed on the corresponding signals.
where, the channel column vector and the transmit signal corresponding to the FE signal are moved to the front column and row, respectively. Then, whereas the conventional LLL starts from the first column, the proposed algorithm skips the first column and performs the LLL processing only on H ^{′}. In other words, the initial value of the column index k is changed from 2 to 3. The output of the proposed algorithm is a 4×4 transformation matrix \(\bar {\mathbf {T}}\) whose rightlower submatrix is the same as T ^{′}, which means that \(\bar {\mathbf {T}} = \left [ \begin {array}{cc} 1 & \mathbf {0}_{1 \times 3} \\ \mathbf {0}_{3 \times 1} & \mathbf {T}' \end {array} \right ].\)
Here, we perform the partial SIC during the LLL processing to obtain the signal s ^{′} which is used to compute the RA in (17). There might exist some false ETs, because the RA is computed with a signal which is obtained by not the original FSD but the SIC on the SE stage. This leads to a tradeoff between the BER performance and the computational complexity as a function of the parameter A in (17). However, the proposed algorithm reduces the complexity considerably with little performance degradation, which is shown in the next Section.
3.2.1 Input channel matrix conversion (line 1 in Table 2)
First, the column vectors of H corresponding to the FE stage and SE stage are exchanged, resulting in a converted channel matrix \(\bar {\mathbf {H}}\). In this way, signals corresponding to the SE stage are displaced to the upper level in the tree structure so that partial SIC detection can be applied to the corresponding signals without considering the parent nodes during the FE stage.
3.2.2 Modification of the index for LR processing
The LRaided FSD priorly eliminates FE signals and performs LRaided SIC detection during the SE stage. This means that column vectors corresponding to the SE stage are those that actually require the basisreduction process. Hence, as detailed in the initialization step and line 21 in Table 2, the column search index for LR processing, k, is modified from [2,⋯,N _{ T }] to [2+N _{ p },⋯,N _{ T }].
3.2.3 ET check (lines 4–17 in Table 2)
Prior to each LR iteration, an ET check is performed by adopting the RA criterion, which is computed by the symbol partially detected with the intermediate latticereduced channel. If the detected symbol is within the predetermined boundary, LR processing is terminated. Note that this operation is performed only when column swapping occurs so that the channel condition is modified. Furthermore, the symbol index where partial detection is performed is determined according to the columnswapping index, as shown in line 23 in Table 2.
3.2.4 Modified QRD (line 2 in Table 2)
The LRaided FSD uses the nulled channel matrix, H ^{′} in (16), which consists of the column vectors of the channel matrix corresponding to the SE stage, as the input of the CLLL process. Meanwhile, the proposed algorithm uses the full channel matrix H after converting to \(\bar {\mathbf {H}}\). Let H ^{′}=Q ^{′} R ^{′} and \(\bar {\mathbf {H}} = \bar {\mathbf {Q}} \bar {\mathbf {R}}\), which are the results of the QRD. Then, the output matrices (Q ^{′},R ^{′}) and (\(\bar {\mathbf {Q}}, \bar {\mathbf {R}}\)) differ from each other. This means that the proposed algorithm performs LR in an undesired manner, as LR processing is dependent on the QRD result.
The average power of the diagonal terms of R ^{′}, the leftmost terms of \(\bar {\mathbf {R}}\), and the fraction of the leftmost terms normalized by the diagonal terms on each tree level. The channel condition is given by the Rayleigh fading channel ordered by the FSD channel ordering
Tree level  Diagonal terms  Leftmost terms  Fraction 

1  4.1858  0.0879  0.0210 
2  3.7763  0.1615  0.0428 
3  4.0852  0.2603  0.0367 
4  4.5849  0.3727  0.0813 
5  5.0649  0.5306  0.1048 
6  5.6178  0.7442  0.1325 
7  6.3230  1.4036  0.2220 
Average  4.8054  0.5087  0.1059 
4 Simulation results
In this section, the BER performance and the computational complexity of the conventional CLLL is compared to that of the proposed CLLL with ET for LRaided FSD through computer simulations. The simulation was conducted on an uncoded 8×8 MIMO system with 256QAM, with N _{ p }=1 for LRaided FSD. Here, E _{ b } denotes the average energy per information bit arriving at the receiver. Thus, the signaltonoiseratio (SNR) is given by \(E_{b} / N_{o} = N_{R} / \left (\log _{2} (\Omega ) {\sigma _{n}^{2}}\right)\).

The multiplication of l×m and m×n real (complex) matrices requires 2 l m n (8 l m n) FLOPs.

The Moore–Penrose pseudoinverse of an m×n real matrix requires 2m ^{3}−2m ^{2}+m+16m n FLOPs.
5 Conclusions
In this paper, we proposed a lowcomplexity LR algorithm with ET for detectors with tree searching. Whereas the ILR [25] is limited to LRaided SIC detectors, the proposed algorithm is an extension of the conventional approach to LR for detectors with tree searching. We performed an additional hypothetical analysis and several novel modifications to the conventional algorithm in order to overcome its limitations.
We verified the performance of the proposed algorithm with a computer simulation, demonstrating that the average number of column swaps was reduced, with negligible BER performance degradation. Furthermore, for a fair comparison, the computational complexity was analysed in terms of the number of FLOPs, indicating a significant reduction in computational complexity.
Declarations
Acknowledgments
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF2015R1A2A2A01004883).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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